Problem 21.2 Hints
Problem 21.2
Step 1: Show that the problem reduces to
� ��
�
� � �
1
Γ1
Δ=
.
inf �Δ�∞ :
0
Γ�2
Δ∈Rn
(1)
We have made this kind of argument several times.
Step 2: Develop a geometrical interpretation of this minimization, based on
the intersection of sets
�
�
Bβ = Δ ∈ Rn : �Δ�∞ = max |δi | ≤ β
a n-d box
i
and
L=
�
� � �
��
�
Γ1
1
Δ ∈ Rn :
Δ
=
0
Γ�2
a (n − 2)-d affine subspace.
How large must β be for some Δ to be a member of both sets? How small can β
be? Where in/on the box Bβ ∗ is an “optimal” Δ going to appear (think about
R3 , where Bβ is a cube and L is a line).
Step 3: We need to simplify the problem. Relax one dimension (say j) of the
problem. That is, let
Δ−j = (δ1 , . . . , δj−1 , δj+1 , . . . , δn ) .
(omiting δj ). Consider the relaxed problem (verify it is a relaxation?):
�
� � �
� ��
Γ1
1
Δ
=
.
inf βj : �Δ−j �∞ ≤ βj ,
Γ�2
0
What is the geometric interpretation here? There is a simple way to “remove”
δj also from the linear constraint (by substitution). What remains is a problem
that looks like our original rank-1 µ problem, but where all quantities are real.
Use this formulation to derive a closed form for βj . Consider this the “crux” of
the problem, and make sure to include it in your solution. The answer should
be in terms of the elements of Γ1 and Γ2 .
Step 4: Recall this is a relaxed problem; when will the solution Δ correspond
to a point in the box Bβj (give a checkable condition)? In this case, then βj is
indeed the solution to (1). Argue that some dimension must produce a feasible
solution. Which of the βj must necessarily be the feasible one (that is, will it
be large or small)?
1
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6.241J / 16.338J Dynamic Systems and Control
Spring 2011
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